For a meta-analysis, means and s.d. for the experimental and control groups were required to calculate the mean difference (MD or yi) and variance (vi) of each study with the escalc function of metafor.
In some studies, s.d. needed to be back-calculated from reported p-value and n in experimental (ne) and control (nc) groups (no dispersion measures available).
A problem arise when MD tends toward zero or is null (p-value is always not significant). In such cases, back-calculated s.d. also tends toward zero because MD is the numerator in the formula (see http://handbook.cochrane.org/chapter_7/7_7_3_3_obtaining_standard_deviations_from_standard_errors.htm)
Indeed, very small or null s.d. are not valid when p-value is not significant. Moreover, inclusion of a study with null s.d. in a dataset leads to a warning message in rma.mv: there are outcomes with non-positive sampling variances; and the model doesn't solve.
My question: one approach would be to exclude all studies where back-calculated s.d. is very small or null due to small or null MD. However, these not significant studies contribute to the overall knowledge and should be included somehow in the meta-analysis.
Or should a s.d. be imputed to these studies; e.g., the mean s.d. of the other not significant studies where s.d. could be computed from a dispersion measure (e.g., sem or sed).
Or something else?